The $N/D$ study on the singularity structure of $\pi N$ scattering amplitudes

TL;DR

This paper employs the $N/D$ method to analyze the singularity structure of $\pi N$ scattering amplitudes, revealing the existence of virtual states and the essential singularity at the $u$ channel cut endpoint, independent of input details.

Contribution

It demonstrates that the subthreshold resonance $N^*(890)$ existence is input-independent and identifies virtual states induced by $u$ channel exchanges, clarifying the singularity structure.

Findings

Existence of virtual states induced by $u$ channel nucleon exchanges.

The $N^*(890)$ resonance existence is independent of dynamical input.

The end point of the $u$ channel segment cut is an essential singularity.

Abstract

The $N/D$ method is used to study the $S_{11}$ channel low energy $\pi N$ scattering amplitude. The input of left cuts are obtained from various phenomenological models. With the aid of the production representation, the total phase shifts can be decomposed into different contributions, and it further reveals that the existence of subthreshold resonance $N^*(890)$ doesn't depend on the details of the dynamical input. Additionally, it is found that there exist virtual states in partial waves, which are induced by the $u$ channel nucleon exchanges. These virtual states accumulate at the end point of the $u$ channel segment cut. The end point is hence the essential singularity of the full amplitude on the second sheet of complex $s$ plane.